Comparing dominance of tennis’ big three

via multiple-output Bayesian quantile regression models

Bruno Santos

University of Kent

Introduction

Big Three

  • Roger Federer

  • Rafael Nadal

  • Novak Djokovic


- Won 63 out of 77 Grand Slam tournaments, between Wimbledon in 2003 until 2022.

Dominance

  • List of Grand Slam Winners:
January-2022
Player Titles
1. Roger Federer 20
1. Rafael Nadal 20
1. Novak Djokovic 20
4. Pete Sampras 14
5. Roy Emerson 12
Currently, August-2022
Player Titles
1. Rafael Nadal 22
2. Novak Djokovic 21
3. Roger Federer 20
4. Pete Sampras 14
5. Roy Emerson 12


Question: Who is more dominant between the Big Three?

How to measure dominance in a tennis match

Important notes:

  • A tennis match is divided into sets and games.

  • A player with most sets wins the match.

  • A player can win more games, but still lose the match.

    • Example: 7-6, 0-6, 7-6.

  • Solution:

    • Relative points: ratio points won/lost in a match.

    • Duration of the match.

Data

  • Data organised by Jeff Sackmann in the repository:

  • All matches from the Big Three, between 1998 and the US Open in 2021.

    • Excluding Davis Cup and Olympic Games matches.

    • Also matches played on carpet.

  • We should condition on some variables:

    • type of tournament (Grand Slam, Masters 1000, …);
    • surface (clay, grass and hard courts);
    • wins and losses;
    • rank of opponent.

Data distribution

Bayesian quantile regression for multiple output response variables

Directional quantile regression model

  • Response variable is defined as \({\boldsymbol Y} \in \mathbb{R}^k\).

  • Directional index can be defined by \({\boldsymbol \tau} \in \mathcal B^k := \{ {\boldsymbol v} \in \mathbb{R}^k: 0 < || {\boldsymbol v} || < 1. \}\).

    • \({\boldsymbol \tau} = \tau {\boldsymbol u}, \tau \in (0,1)\).
    • Direction: \({\boldsymbol u} \in \mathcal{S}^{k-1} := \{ z \in \mathbb{R}^k : ||z|| = 1 \}\);

  • Define \(\boldsymbol \Gamma_u\), an arbitrary \(k \times (k-1)\) matrix of unit vectors.

    • \((u \, \vdots \, \Gamma_u)\) is an orthonormal basis of \(\mathbb{R}^k\).

DEFINITION:

The \(\boldsymbol \tau\)th quantile of \(\boldsymbol Y\) is the \(\tau\)th quantile hyperplane obtained from the regression:

  • \({\boldsymbol Y_u} := {\boldsymbol u}^{'} {\boldsymbol Y}\) on the marginals of \({\boldsymbol Y}^{\perp} := {\boldsymbol \Gamma_u}^{'} \boldsymbol Y\) with an intercept term.

Estimation setup

The \(\boldsymbol \tau\)th quantile of \(\boldsymbol Y\) is any element of the collection \(\Lambda_\tau\) of hyperplanes

\[\lambda_\tau := \{ \boldsymbol y \in \mathbb{R}^k : \boldsymbol u^{'} \boldsymbol y = \hat{\boldsymbol b}_\tau \boldsymbol \Gamma^{'}_u \boldsymbol y + \hat{a}_\tau \},\]

such that \((\hat{a}_\tau, \hat{\boldsymbol b}_\tau)\) are the solutions of the minimization problem

\[\min_{(a_\tau, \boldsymbol b_\tau) \in \mathbb{R}^k} E[\rho_\tau(\boldsymbol u^{'} \boldsymbol y - \boldsymbol b_\tau \boldsymbol \Gamma^{'}_u \boldsymbol y - a_\tau )].\]

where \(\rho_\tau(u)\) is a known loss function in the quantile regression literature defined as

\[\rho_\tau (u) = u(\tau-\mathbb{I}(u<0)), \quad 0 < \tau < 1.\]

Upper and lower halfspaces



With predictor variables, we have \[\lambda_\tau(\boldsymbol X) = \{ \boldsymbol u^{'} \boldsymbol y = \hat{\boldsymbol b}_\tau \boldsymbol \Gamma^{'}_u \boldsymbol y + \boldsymbol x^{'} \hat{\boldsymbol \beta}_\tau + \hat{a}_\tau \},\]

We can say that each element \((\hat{a}_\tau, \hat{\boldsymbol b}_\tau, \hat{\boldsymbol \beta}_\tau)\) define an upper closed quantile halfspace \[\begin{equation*} H^+_{\tau \boldsymbol u} = H^+_{\tau \boldsymbol u} (\hat{a}_\tau, \hat{\boldsymbol b}_\tau, \hat{\boldsymbol \beta}_\tau) = \{ \boldsymbol y \in \mathbb{R}^k : \boldsymbol u^{'} \boldsymbol y \geq \hat{\boldsymbol b}_\tau \boldsymbol \Gamma^{'}_u \boldsymbol y + \boldsymbol x^{'} \hat{\boldsymbol \beta}_\tau + \hat{a}_\tau \} \end{equation*}\] and an analogous lower open quantile halfspace switching \(\geq\) for \(<\).



Properties

  • Probabilistic nature of quantiles:

\[P(\boldsymbol Y \in H_{\tau \boldsymbol u}^-) = \tau,\]

Quantile region

Moreover, fixing \(\tau\) we are able to define the \(\tau\) quantile region \(R(\tau)\) as \[\begin{equation*} R(\tau) = \bigcap_{{\boldsymbol u} \in \mathcal{S}^{k-1}} H_{\tau \boldsymbol u}^+. \end{equation*}\]

Bayesian directional quantile regression model

Consider the mixture representation of the asymmetric Laplace distribution

\[\begin{align*} Y_i | w_i &\sim N(\mu + \theta w_i, \psi^2 \sigma w_i) \\ w_i &\sim \mbox{Exp}(\sigma) \\ &\Updownarrow \\ Y &\sim AL(\mu, \sigma, \tau) \\ \end{align*}\]

Then one can consider that, for each direction \(u\), \[Y_u | \boldsymbol b_\tau, \boldsymbol \beta_\tau, \sigma, w \sim N(Y^\perp b_\tau + \boldsymbol x^{'} \boldsymbol \beta_\tau + \theta w_i, \psi^2 \sigma w_i),\]

Application results

Model choices

  • \(Y_1\): Relative points won.

  • \(Y_2\): Minutes played.

  • Covariates:

    • Player (Federer, Nadal, Djokovic);
    • Surface;
    • Win or loss;
    • Type of tournament;
    • Top 20 player opponent or not;

  • For the model, we fix \(\tau = 0.25\) and consider 180 directions in the unit circle.

  • We consider interaction effects between player and the other covariates.

Effect of win and losses

Effect of tournament

Effect of Top 20

Effect of surface

Final discussion

Conclusions

  • This model does not need to make any probability assumptions in order to reach its conclusions.

  • Nadal’s dominance in clay courts is unmatched.

  • Federer dominance in grass courts is also visible.

  • The same way as Djokovic dominance in hard courts.

  • In the time dimension, Federer shows an edge during wins.

  • For most comparisons, Djokovic seems the most dominant player.





Thank you!

  • b.santos@kent.ac.uk

COMPSTAT 2022